Envy-free matchings in bipartite graphs and their applications to fair division

Guarantees in Fair Division: General or Monotone Preferences

“…Kuhn's 1967 [22] n person generalisation of Divide and Choose promptly implements the minM ax guarantee in our model (Theorem 1). Except for a recent discussion in Aigner-Horev and Segal-Halevi [1] for additive utilities D&C n has not received much attention, a situation which our paper may help to correct. In particular, unlike the Diminishing Share (Steinhaus [35]) Moving Knife (Dubins and Spanier [20]), and Bid and Choose rules, it is very well suited to divide mixed manna, i. e., containing subjectively good and bad parts, as when we divide the assets and liabilities of a dissolving partnership.…”

“…It is clear that EP n (S; u) is non empty if u is additive. Let B[S] be the set of shares included in S: Lyapunov Theorem implies that the range u(B[S]) is convex, so it contains 1 n u(S); then we replace n by n − 1 and repeat the argument on the remaining share.…”

“…= (1, 1). For the Leontief utility u(x, y} = min{x, y} we have M axmin(u; 2) = 1 2 at the equal split partition, while minM ax(u; 2) = 0 at the equipartition {(1, 0); (0, 1)}; for the anti-Leontief utility v(x, y) = max{x, y} the same two equipartitions give dually minM ax(v; 2) = 1 2 and M axmin(v; 2) = 1. Clearly the utility profile (M axmin(u; 2), M axmin(u; 2)) is not feasible.…”

“…In D the inequality M axmin(u; n) ≤ u( 1 n !) is also true 1 , and it is feasible to give an 1 Pick a hyperplane H supporting the upper contour of u at 1 n ! ; the lower contour of u at 1 n !…”

“…is also true 1 , and it is feasible to give an 1 Pick a hyperplane H supporting the upper contour of u at 1 n ! ; the lower contour of u at 1 n ! contains one closed half-space cut by H, and every division of !…”